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The only train service that directly connects Kansai International Airport with Kyoto is the limited express train, "Haruka" provided by JR. It takes about 75 minutes and costs 3,290 yen one way (fares can change depending on the day of the week). The train terminates in JR Kyoto Station, the main portal for Kyoto City.
To reach the "Haruka" platform, exit the airport from exit D or E on the second floor and cross the concourse to enter the JR Kansai Airport Station.

There are discounted program for Foreign tourists to purchase must have Foreign passport.
http://www.westjr.co.jp/global/en/travel-information/pass/icoca-haruka/

Here is a brief introduction to the train systems in Japan. Train services in Japan are divided into three categories: JR, private railways and subways. JR provides services throughout Japan and runs the train directly connecting Kansai International Airport and Kyoto. The major private railways mainly run between large cities and there is no direct private railway service between Kyoto and the airport. Subway lines run within a city, for example, Kyoto and Osaka, respectively.

A number of other train services connect the airport and Kyoto via Osaka. Either the JR Kanku Express or the private Nankai line will take passengers as far as Osaka. From Osaka, passengers can proceed on to Kyoto using any of following railway lines depending on one's destination: JR, Hankyu or the Keihan line.
"Kansai Thru Pass" is a convenient unlimited rides ticket applicable to about 40 transportation companies. The 2-day pass and 3-day pass are available. Showing this pass will give you a discounts or other privileges at many places around the applicable area.
http://www.surutto.com/tickets/kansai_thru_english.html

The airport limousine bus provides a direct connection between Kansai International Airport and Kyoto. It takes approximately 90 minutes and the fare is 2,500 yen one way. The bus terminates at the JR Kyoto Station Hachijo Exit and departs from Platform #8, on the first floor of the airport terminal building near Exit D.
http://www.kate.co.jp/pc/index_e.html

The shared taxi service is a very convenient means of transport taking passengers directly from the airport to their final destination in Kyoto City. There are two taxi companies offering this service and both charge 3,500 yen per person. Discounts are available depending on the number of people boarding and also for a round-trip ticket.
Please note that a reservation is necessary and some areas of Kyoto City are outside the service area.
Yasaka Taxi's Kansai Airport Shuttle is a convenient and comfortable taxi service between Kansai International Airport and Kyoto city area (3,500 yen/person; one way; including tax).
(Free of charge for one large suitcase per person. Discount is available for Internet reservation, round-trip reservation and group reservation)
Reserved shuttle service is also available for max 9 people/vehicle (25,000 yen/vehicle between Kyoto and Kansai International Airport; one way; including tax)
http://www.yasaka.jp/english/